{cal 3 exm3} cylindricalr and spherical cords

Quiz!

$$S(u,v)=<u+2v,2v-u>$$

1. Jacobian?
   1. $\partial u<1,-1>\partial v<2,2>$
   2. Take $ad-bc=4$
2. What is area of $\sum u,v$? $\pi$ (what is area of unit circle basically)
3. $\int {\int }_{\sum }(1)(1)dS=\int {\int }_{\sum uv}(1)(4)dA={\int }_0^{2\pi }{\int }_0^1(4r)drd\theta$

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Introduction to cylindrical cords!

• $x(r,\theta ,z)=?$
• We just plug in for

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Q1 Spherical

$$S(r,\theta ,\phi )=<r\sin \phi \cos \theta ,r\sin (\phi )\sin \theta ,r\cos \phi >$$

1. What is @param of sphere of rad =3?
   1. Simply take some $L(\theta ,\phi )$ = $S(3,\theta ,\phi )$
   2. Where sphere has the constraints of $[0,2\pi ]\times [0,\pi ]$
2. What is the jacobian?
   1. Take the cross product of both partials for $L_{\theta }\times L_{\phi }=<-9{\sin }^2\phi \cos \theta ,-9{\sin }^2\phi \sin \theta ,-9\sin \phi \cos \phi >$
   2. || cross_product || = $9\sin \phi$
3. Setup iterated $\int$ for the surface area
   1. Finding bounds: $\phi =\left[\frac{\pi }{3},\pi \right],\theta =[0,2\pi ]$
   2. Plugin to $\int$ below then solve
   3. $$\int {\int }_{D}1dS=\int \int 9\sin \phi d\theta d\phi$$

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Went over jacobian of spherical coordinates lamr edu. Be comfy doing smtn like the last problem

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Q1

Given some

$$\int \int {\int }_D{z}^2+4dV$$

as some cylindrical transformation using

dV=r dz \space dr \space d \theta $$ With some inner radius =1, outer =3, z=1, z=4. Hence our bounds is for this:

D=[1,3] \times [0,2\pi] \times[1,4]

$$Plugginginwefinallyobtain:$$

\int^{2\pi}{0} \int{1}^3 z^2+4 dr \space d\theta